Activity 31:

1. Activity 31: Dividing Polynomials (Section 4.2, pp. 325-331)

2. Example 1: Divide 63 by 12.

3. Division Algorithm: If P(x) and D(x) are polynomials, with D(x) ≠ 0, then there exist unique polynomials Q(x) and R(x), where R(x) is either 0 or of degree strictly less than the degree of D(x), such that P(x) = Q(x)D(x) + R(x) The polynomials P(x) and D(x) are called the dividend and divisor, respectively; Q(x) is the quotient and R(x) is the remainder.

4. Example 2: Divide the polynomial P(x) = 2x2 − x − 4 by D(x) = x − 3. 2x + 5 x − 3 2x2 − x − 4 -2x2 + 6x 2x2 – 6x 5x – 4 -5x + 15 5x – 15 11

5. Example 3: Divide the polynomial P(x) = x4 − x3 + 4x + 2 by D(x) = x2 + 3. x2 – x – 3 x4 − x3 + 4x + 2 x2 + 3 –x4 – 3x2 x4 + 3x2 − x3 – 3x2 + 4x + 2 + x3 + 3x – x3 – 3x – 3x2 + 7x + 2 – 3x2 – 9 +3x2 + 9 7x + 11

6. Synthetic Division: Use synthetic division to divide the polynomial P(x) = 2x2 − x − 4 by D(x) = x − 3. root

7. Example 4: Use synthetic division to find the quotient Q(x) and the remainder R(x) when: f(x) = 3x3 + 2x2 − x + 3 is divided by g(x) = x − 4.

8. Example 5: Use synthetic division to find the quotient Q(x) and the remainder R(x) when: f(x) = x5 − 4x3 + x is divided by g(x) = x + 3.

9. Remainder Theorem: If the polynomial P(x) is divided by D(x) = x − c, then becomes Plugging in x=c to the above equation one sees that

10. Example 6: Let P(x) = x3 + 2x2 − 7. (a) Find the quotient and the remainder when P(x) is divided by x + 2. (b) Use the Remainder Theorem to find P(−2).

11. Factor Theorem: The number c is a zero of P(x) if and only if x−c is a factor of P(x); that is, P(x) = Q(x) · (x − c) for some polynomial Q(x). In other words, in Synthetic division the R(x) = 0 that is the last term is zero.

12. Example 7: Use the Factor Theorem to determine whether x + 2 is a factor of f(x) = 3x6 + 2x3 − 176. YES!!!!!!!!!!

13. Example 8: Find a polynomial of degree 3 that has zeros 1, −2, and 3, and in which the coefficient of x2 is 3.